J. Phys. A: Math. Theor.**42**(2009) 214018 (6pp) doi:10.1088/1751-8113/42/21/214018

**Collective excitations in strongly coupled** **ultra-relativistic plasmas**

**Peter Hartmann**^{1,2}**, Gabor J Kalman**^{2}**, Kenneth I Golden**^{3}
**and Zoltan Donko**^{1,2}

1Research Institute for Solid State Physics and Optics of the Hungarian Academy of Sciences, H-1525 Budapest, PO Box 49, Hungary

2Department of Physics, Boston College, Chestnut Hill, MA 02467, USA

3Department of Mathematics and Statistics, College of Engineering and Mathematical Sciences, University of Vermont, Burlington, Vermont 05401-1455, USA

E-mail:kalman@bc.edu

Received 15 October 2008, in final form 19 November 2008 Published 8 May 2009

Online atstacks.iop.org/JPhysA/42/214018
**Abstract**

In the collective mode spectrum of a relativistic, strongly coupled plasma, novel physical effects emerge, which are absent both in the weakly coupled relativistic and in the strongly coupled non-relativistic plasmas. Inspired by the pseudo-relativistic behavior of the electron gas in two-dimensional graphene layers, we address the problem of a classical two-dimensional, ultra- relativistic system of charged particles. We investigate the mode dispersion and damping both through molecular dynamics simulations and analytically via the quasi-localized charge approximation and develop modifications of the theory appropriate for this system. The new aspect introduced in the simulation is the decoupling of particle velocities from the particle momenta. As for new physical features, their origin is to be sought in the constancy of particle speeds and in the broad distribution of ‘plasma frequencies’, mimicking the similar distribution of momenta is causing the system to emulate the behavior of a collection of an infinite number of oscillators. Of particular interest is the strongly reduced damping at weak coupling, brought about by the disappearance of the Landau damping and the greatly enhanced damping at strong coupling, caused by the phase mixing of the coupled plasma oscillators.

We suggest the possible experimental detection of these effects in graphene.

PACS numbers: 52.27.Ny, 52.27.Gr, 52.35.Fp, 73.21.−b

The problem of relativistic plasmas first acquired attention in the works by Russian scientists [1]. The attention then focused on the behavior of plasmas where relativistic effects were due to high temperature. In a different line of research, the effect of Coulomb interaction on the behavior of the degenerate relativistic electron gas, principally in the context of white dwarf

interiors, was studied in [2] and [3]: here relativistic effects are brought about by the high
Fermi energy. In both situations, the Coulomb interaction energy is small compared to the
kinetic energy, and in the usual parlance, the plasma is*weakly coupled.* In the case of an
ultra-relativistic electron gas, when the ultra-relativistic limitpF/mc→ ∞and for the single
particle energyεp =c!

p^{2}+(mc)^{2},εF = pF/c(p_{F}is the Fermi momentum) are reached,
this is, in fact, unavoidable, since the conventional coupling constant rs goes to the limit
e^{2}/¯hc$1/137.

Relativistic effects manifest themselves in two different ways. First, once the particle
velocities become comparable to *c, the speed of light, the instantaneous static interaction*
between the charges has to be complemented by the fuller description through the retarded
Lienard–Wiechert potential or by explicitly introducing photons as carriers of the interaction.

Second, the relativistic particle dynamics has to be described in terms of a ‘variable (momentum-dependent)’ mass.

In this paper we address the behavior of the collective modes, in particular plasma
oscillations under relativistic conditions. In the weak-coupling limit the two effects referred to
the above are easily handled and do not lead to any dramatically new physical phenomena. As
to the first, in the Vlasov (or random-phase-approximation (RPA)) description the interaction
is mediated by the mean field only, which can easily be determined via a full set of Maxwell
equations, as is already done in the conventional computation of the elements of the dielectric
tensorεµν(k, ω)=δµν+αµν(k, ω). As to the second, the relativistically correct expression for
the plasma frequency is obtained by the replacement of the particle mass by the (single-particle
energy)/c^{2}: m⇒η_{p} =!

p^{2}+(mc)^{2}/c. As a result, each particle acquires its own ‘plasma
frequency’ and the system becomes an aggregate of infinitely many components. However,
within the RPA, this feature does not pose any particular problem, since in a multi-component
system the individual polarizabilities simply add to generate the total longitudinal dielectric
functionε(k, ω) = 1 +"

Aα^{A}(k, ω), which is tantamount to replacing 1/min the plasma
frequency by a suitably defined&1/ηp'.

It is known, however, from the study of strongly coupled binary systems [4] that in such systems the simple addition law for the polarizabilities does not hold and the degeneracy of the plasma mode is removed. In fact, the combination of relativistic behavior and strong coupling creates a unique physical system, resembling that of a collection of a large number of coupled oscillators. Thus, in these novel systems the collective excitations are expected to exhibit an unconventional structure.

Recently, a number of plasma-like systems have become of interest, which are not weakly
coupled and exhibit at the same time relativistic or quasi-relativistic behavior. Such systems
are (i) laser-compressed plasmas, where the combination of high temperature and high density
results in ' > 1 [5], (ii) quark–gluon plasmas [6], where the dominant interaction is the
Coulomb-like color field and where experiments seem to indicate a ' > 1 situation and
(iii) two-dimensional graphene layers [7], where the peculiar band structure leads to a
pseudo-relativistic dynamics characterized bym_{eff} =0 andη_{p} =p/c_{eff} withc_{eff} $c/400
andr_{s,eff}$400/137 (on a vacuum substrate).

In this paper, we explore the novel effects in the collective mode structure that arise
in moderately to strongly coupled relativistic plasmas. The model we consider has been
inspired by the electron gas in graphene. We have studied a 2D system of (negatively) charged
classical pseudo-ultra-relativistic massless particles with single-particle energyε** _{p}**=pc

_{eff}, in a neutralizing background. We have investigated the collective mode structure analytically, by invoking the quasi-localized charge approximation (QLCA) (see, e.g., [8]), and we have performed a series of molecular dynamics (MD) simulations on the system over a wide range of coupling values. We contend that the classical approach is likely to provide an adequate

0 0.5 1 1.5 2 2.5 3 3.5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 ω/ω0

ka (a) MD - S(k,ω)

2D OCP RPA eQLCA pQLCA

0 0.5 1 1.5 2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

∆ω/ω0 (FWHM)

ka (b)

Γ= 1 MD - S(k,ω)

pQLCA

0.1 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.4 0.2 0

**Figure 1.** ' =1: (a) dispersion curves for from MD simulation and theoretical approaches;

note that the OCP peak is detectable only up to*ka*=2, while the current one survives for much
higher*ka*values; (b)*S(k,*ω) linewidth part from MD simulation and imaginary part of the pQLCA
dispersion.

0 0.5 1 1.5 2 2.5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 ω/ω0

ka (a) MD - S(k,ω)

RPA QLCA eQLCA pQLCA

0 0.2 0.4 0.6 0.8 1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

∆ω/ω0 (FWHM)

ka (b)

Γ= 10 MD - S(k,ω)

pQLCA

0.1 0 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.4 0.2 0

**Figure 2.** '=10: (a) dispersion curves for from MD simulation and theoretical approaches;

(b)*S(k,*ω) linewidth from MD simulation and imaginary part of the pQLCA dispersion.

description for the collective mode structure of the degenerate quantum system as well; this
contention is supported by the results of similar studies on a variety of systems [9]. The
coupling constant for the 2D ultra-relativistic classical plasma and for the degenerate electron
gas are defined, respectively, as' = βe^{2}/a, r_{s} = e^{2}/¯hc (chere and in the sequel stands
for the effective ‘light speed’; *a* is the Wigner–Seitz radius,π na^{2} = 1). Then, based on
equating the average kinetic energies in the two systems, the correspondence'⇒(3/2)√grs

(gis the spin/valley multiplicity factor) can be established. The MD simulations are based on the PPPM technique [10]; particles are confined in a 2D layer with periodic boundary conditions. From the point of view of simulations, the most important new aspect is that momentum and velocity decouple from each other: the momentum satisfies the equation of motion, but the velocity determines the displacement. The absolute value of the velocity remains constant and interactions change its direction only.

Results of the MD simulation for the dispersion and damping of the plasmon mode for ' =1,10 and 100 are displayed in figures1–3, respectively. The dispersion is determined

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 ω/ω0

ka (a)

MD - S(k,ω) 2D OCP - L QLCA pQLCA

0 0.2 0.4 0.6 0.8 1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

∆ω/ω0 (FWHM)

ka

(b) MD - S(k,ω) Γ= 100

pQLCA

0.1 0 0.2 0.3 0.4 0.5 0.6 0.7

0.4 0.2 0

**Figure 3.** ' = 100: (a) dispersion curves from MD simulation and theoretical approaches;

(b)*S(k,*ω) linewidth from MD simulation and imaginary part of the pQLCA dispersion.

0 0.2 0.4 0.6 0.8 1 1.2 1.4

1 10 100

∆ω/ω0 (FWHM)

Γ

ka = 2.0

MD - S(k,ω) 2D OCP

**Figure 4.***S(k,*ω) linewidth from MD simulations versus'. Compared are the results for classical
2D OCP and the present ultra-relativistic system at the wave number*ka*=2.0.

from the peak positions of the dynamical structure function *S(k,* ω), while the damping
is characterized by the width of the respective peaks. For comparison, the corresponding
results for the non-relativistic strongly coupled one-component plasma (OCP) are also shown.

Figure4 contrasts the damping as a function of' of the relativistic versus non-relativistic systems.

In order to describe the observed features various theoretical models have been explored.

As a starting point, the classical longitudinal response function χ (k, ω) has been derived
within the framework of the conventional Vlasov (RPA) description (this is the classical
equivalent of the approach followed by Das Sarma*et al*[11]) and is used to generate the
dispersion. It is important to realize that the phase velocity of the plasmon is always higher
than*c*and consequently there is no Landau damping. The absence of the Landau damping is
manifested in the structure ofχ (k, ω)by its becoming purely real: remarkably, the absence
of an imaginary part inχ (k, ω)allows one to express the dispersion relation explicitly (see
equation (3)). The RPA description should be appropriate for'!1. For higher'values the

QLCA theory, where the frequency is expressed in terms of the dynamical matrix*D(k),*
D(k)=1

k¯{+(0)−+(k)}

(1) +(k)= 1

2π

#

d^{2}rψ (¯r)¯ e^{−}^{ik}^{·}** ^{r}**h(r) ψ (r)= 1

r^{3}(3 cos^{2}ϑ−1)

(h(r)is the pair correlation function, ¯r=r/aand cosϑ= ** ^{k}**kr

^{·}

**) should be applicable. However, in contrast to the non-relativistic plasma, where it can be argued that for high enough coupling the effect of thermal velocities is negligible, here the**

^{r}*c*=constant condition makes the validity of such an approximation doubtful. Thus, we have applied the ‘extended’ version of the QLCA, the eQLCA theory [12], where the random motion is accounted for by combining the RPAχ (k, ω)with the dynamical matrixD(k). The resulting dispersion relation is

.(k)
ω_{0} =!

k¯ 1 +_{2'}^{k}^{¯} +D(k)

$1 +_{4'}^{¯}^{k} +D(k)

k¯=ka
ω^{2}_{0} = 2π ne^{2}

a&p/c' =π ne^{2}c^{2}β/a=π nc^{2}'. (2)
It can be noted that equation (2) incorporates both the RPA and the ‘classical’ QLCA dispersion
relations: the former is obtained by settingD(k)=0 and the latter by letting'→ ∞.

For sufficiently strong coupling, the physical effect that we have referred to as the system
becoming a continuous distribution of coupled harmonic oscillators comes into play. Starting
with the description of the system via a multi-component generalization of the QLCA [4], the
model (to be referred to as pQLCA) leads to a set of*N*(=number of particles) coupled equations
in momentum space; by converting the sum into an integral over the thermal distribution of
momenta, an integral equation for.(k)is obtained:

−/(k) /(0)

# _{∞}

0

e^{−}^{x}

1−^{[.(k)]}/(0)^{2}xxdx =1

/(k)=ω_{0}^{2}{k¯++(k)}
0(r)= 1

r^{3}(3 cos^{2}ϑ−1). (3)
The mode frequency, .(k), that emerges as a solution is a complex quantity, indicating
damped oscillations. Since the classical RPA is devoid of any damping mechanism, the origin
of the damping here must be sought in the phase mixing of the distributed oscillators (cf [13]).

The dispersion curves resulting from the various approaches are displayed in figures1–3 (the insets illustrate the behavior at low wave numbers).

The conclusions derived from the comparison between the theoretical predictions and the MD simulations can be summarized as follows:

(1) The low coupling ' = 1. Figure 1 reveals that even at this low-coupling value the MD results deviate quite substantially from the RPA prediction, while the corresponding results for the non-relativistic (NR) OCP do not. This seems to indicate that strong- coupling effects are more pronounced in the relativistic than in the NR case. It can also be noted that the eQLCA provides an improvement over the RPA, coming closer to the MD points.

(2) The medium coupling'=10. Figure2points at the complete inadequacy of the RPA in
this domain. The best fit seems to be provided by the pQLCA; for small*k-values, the MD*
results are below the QLCA and so are the pQLCA values, but with too much depression.

Otherwise, the difference between the QLCA and the pQLCA is not substantial. Contrary to expectations, the eQLCA does not seem to provide an improvement at this'value.

(3) The high coupling' =100 case (figure3) further highlights (cf'=1) the difference between the NR OCP and the relativistic system: this latter seems to be consistently below the NR OCP dispersion. It is the pQLCA that seems to best capture the qualitative features of the dispersion.

(4) Looking at the damping in figure 4, the difference between the '-dependences of the
NR OCP and the relativistic system is striking. The absence of the Landau damping in
the latter should be responsible for the much lower damping in the relativistic plasma at
low'values. As to the high'domain, we believe that the broad phonon spectrum and
the phase mixing of the distributed oscillators (cf above) is responsible for the enhanced
damping in the relativistic plasma. This seems to be the explanation also for the fact
that there is a non-vanishing damping even in the absence of Landau damping at any'
value, as shown in the MD results of figures1–3, which for small*k*values are reasonably
matched by the pQLCA finding.

Commenting finally on the possible relevance of our results to graphene, we have already
argued that the classical approach should render a reasonable description of the collective
excitations even in the degenerate quantum situation. While the graphene rs values are
relatively modest (c=8×10^{5} cm s^{−}^{1},rs =2.74 corresponds to'=8.22), the behavior
portrayed for'=1 and'=10 indicates that while strong-coupling effects would not be
dramatic, they may be detectable by comparing future observations on graphene plasmon
dispersion and damping with predictions of the RPA and of the current theory.

**Acknowledgments**

GJK wishes to thank Constantin Andronescu for his help in the mathematical manipulations.

This work has been supported by the National Science Foundation under grants nos PHY- 0813153, PHY-0514618, PHY-0812956 and PHY-0514619, and by the Hungarian Fund for Scientific Research and the Hungarian Academy of Sciences, OTKA-T-48389, OTKA-IN- 69892, MTA-NSF-102, OTKA-PD-75113. This paper has also been supported by the Janos Bolyai Research Scholarship of the Hungarian Academy of Sciences.

**References**

[1] Klimontovich Yu L 1960*Sov. Phys.—JETP***10**524
Klimontovich Yu L 1960*Sov. Phys.—JETP***11**876
Silin V P 1960*Sov. Phys.—JETP***11**1136
Silin V P 1960*Sov. Phys.—JETP***13**430

Silin V P and Fetisov E P 1962*Sov. Phys.—JETP***14**115

Trubnikov B A 1960 Thesis, translation in AEC-tr-4073 (Oak Ridge, TN: US Atomic Energy Commission)
[2] Jancovici B 1962*Nuovo Cimento***25**428

[3] Akhiezer I A and Peleteminskii S V 1960*Sov. Phys.—JETP***11**1316
[4] Kalman G and Golden K I 1990*Phys. Rev.*A**41**5516

[5] Streitz F H, Graziani F R, More R M, Murillo M S, Surh M P, Benedict L X, Hau-Riege S P, Langdon A B and
London R A 2008 Particle simulations of hot dense matter with radiation*SCCS2008 Book of Abstracts*p 62
[6] Heinz U 2009*J. Phys. A: Math. Theor.***42**214003

Thoma M 2009*J. Phys. A: Math. Theor.***42**214004

[7] Katsnelson M 2008 Graphene: New bridge between condensed matter and quantum electrodynamics*SCCS*
*Book of Abstracts*p 26

[8] Golden K I and Kalman G 2000*Phys. Plasmas***7**14

[9] Kalman G J, Hartmann P, Donko Z and Golden K I 2007*Phys. Rev. Lett.***98**236801
Golden K I, Kalman G J, Donko Z and Hartmann P 2008*Phys. Rev.***B78**045304

[10] Hockney R W and Eastwood J W 1981*Computer Simulation Using Particles*(New York: McGraw-Hill)
[11] Das Sarma S, Hu B Y- K, Hwang E H and Tse W-K B arXiv:0708.3239

Hwang E H and Das Sarma S 2007*Phys. Rev.*B**75**121406(R)
[12] Golden K I, Mahassen H and Kalman G J 2004*Phys. Rev.*E**70**026408

Kalman G J, Golden K I, Donko Z and Hartmann P 2005*J. Phys.: Conf. Series***11**254
Golden K I, Kalman G J, Donko Z and Hartmann P 2008*Phys. Rev.*B**78**045304
[13] Elliott R J, Krumhansi J A and Leath P L 1974*Rev. Mod. Phys.***46**456